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Serre weight conjectures for $\mathrm{GSp}_4$

Daniel Le, Bao V. Le Hung, Heejong Lee

TL;DR

This work proves the weight part of Serre's conjecture for Galois representations valued in $GSp_4$ under tame ramification and explicit $9$-genericity at places above $p$, assuming automorphy and Taylor–Wiles conditions. A novel method avoids needing full knowledge of patched-module supports by propagating weight information through outer weights and common components, leveraging the Emerton–Gee stack and local models to control generizations. The results yield a modularity lifting theorem for $GSp_4$ with small Hodge–Tate weights and generic tame inertial types, and provide explicit Breuil–Mézard cycles that connect local deformation rings to global patched modules. The approach combines étale and coherent patching, local-model geometry, and weight-elimination techniques to establish a robust framework for determining Serre weights and automorphy in the $GSp_4$ setting, with broad implications for Hilbert–Siegel modular forms and related automorphic representations.

Abstract

We prove the weight part of Serre's conjecture for Galois representations valued in $\mathrm{GSp}_4$ that are tamely ramified with explicit genericity at places above $p$ as conjectured by Herzig--Tilouine and Gee--Herzig--Savitt. This improves on arXiv:2304.13879 where an inexplicit genericity hypothesis was required. As an application, we prove a modularity lifting theorem for $\mathrm{GSp}_4$ under similar assumptions.

Serre weight conjectures for $\mathrm{GSp}_4$

TL;DR

This work proves the weight part of Serre's conjecture for Galois representations valued in under tame ramification and explicit -genericity at places above , assuming automorphy and Taylor–Wiles conditions. A novel method avoids needing full knowledge of patched-module supports by propagating weight information through outer weights and common components, leveraging the Emerton–Gee stack and local models to control generizations. The results yield a modularity lifting theorem for with small Hodge–Tate weights and generic tame inertial types, and provide explicit Breuil–Mézard cycles that connect local deformation rings to global patched modules. The approach combines étale and coherent patching, local-model geometry, and weight-elimination techniques to establish a robust framework for determining Serre weights and automorphy in the setting, with broad implications for Hilbert–Siegel modular forms and related automorphic representations.

Abstract

We prove the weight part of Serre's conjecture for Galois representations valued in that are tamely ramified with explicit genericity at places above as conjectured by Herzig--Tilouine and Gee--Herzig--Savitt. This improves on arXiv:2304.13879 where an inexplicit genericity hypothesis was required. As an application, we prove a modularity lifting theorem for under similar assumptions.

Paper Structure

This paper contains 34 sections, 34 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. The main result
  3. Previous methods
  4. Methods
  5. Application
  6. Acknowledgements
  7. Notation
  8. Reductive groups
  9. The group $\mathrm{GSp}_4$
  10. Galois Theory
  11. Miscellaneous
  12. Representation theory
  13. Extended affine Weyl groups
  14. Parabolic shapes
  15. Types and weights
  16. ...and 19 more sections

Key Result

Theorem 1.1.1

Suppose that $\overline{r}$ is potentially diagonalizably automorphic and satisfies the Taylor--Wiles conditions. If $\overline{r}$ is tamely ramified and $9$-generic at all places above $p$, then $W(\overline{r}) = W^?(\overline{r})$.

Theorems & Definitions (82)

  • Theorem 1.1.1: Theorem \ref{['thm:SWC']}
  • Remark 1.1.2
  • Theorem 1.4.1: Theorem \ref{['thm:MLT']}
  • Definition 2.1.1
  • Definition 2.1.2
  • Definition 2.1.3
  • Definition 2.1.4
  • Definition 2.1.5
  • Lemma 2.1.6
  • proof
  • ...and 72 more